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Pole Mapping with Optimal Zeros

We saw in the preceding sections that both the impulse-invariant and the matched-$ z$ transformations map poles from the left-half $ s$ plane to the interior of the unit circle in the $ z$ plane via

$\displaystyle z_i = e^{s_i T} \protect$ (9.8)

where $ s_i$ is the location of the $ i$th pole in the $ s$ plane (assumed to lie in the strip $\vert\mbox{im\ensuremath{\left\{s\right\}}}\vert<\pi/T$ to avoid aliasing). The zeros, on the other hand, were different because the impulse-invariant method started with the partial fraction expansion while the matched-$ z$ transformation started with the factored form of the transfer function.

Therefore, an obvious generalization is to map the poles according to Eq.$ \,$(8.8), but compute the zeros in some optimal way, such as by Prony's method [449, p. 393],[273,297].

It is hard to do better Eq.$ \,$(8.8) as a pole mapping from $ s$ to $ z$, when aliasing is avoided, because it preserves both the resonance frequency and bandwidth for a complex pole [449]. Therefore, good practical modeling results can be obtained by optimizing the zeros (residues) to achieve audio criteria given these fixed poles. Alternatively, only the least-damped poles need be constrained in this way, e.g., to fix and preserve the most important resonances of a stringed-instrument body or acoustic space.

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Modal Expansion
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Matched Z Transformation